/// <summary>
/// Returns 2 raised to the specified power.
/// Provides at least 6 decimals of accuracy.
/// </summary>
internal static FP Pow2(FP x)
{
if (x.RawValue == 0)
{
return(FP.One);
}
// Avoid negative arguments by exploiting that exp(-x) = 1/exp(x).
bool neg = x.RawValue < 0;
if (neg)
{
x = -x;
}
if (x == FP.One)
{
return(neg ? FP.One / (FP)2 : (FP)2);
}
if (x >= FP.Log2Max)
{
return(neg ? FP.One / FP.MaxValue : FP.MaxValue);
}
if (x <= FP.Log2Min)
{
return(neg ? FP.MaxValue : FP.Zero);
}
/* The algorithm is based on the power series for exp(x):
* http://en.wikipedia.org/wiki/Exponential_function#Formal_definition
*
* From term n, we get term n+1 by multiplying with x/n.
* When the sum term drops to zero, we can stop summing.
*/
int integerPart = (int)Floor(x);
// Take fractional part of exponent
x = FP.FromRaw(x.RawValue & 0x00000000FFFFFFFF);
var result = FP.One;
var term = FP.One;
int i = 1;
while (term.RawValue != 0)
{
term = FP.FastMul(FP.FastMul(x, term), FP.Ln2) / (FP)i;
result += term;
i++;
}
result = FP.FromRaw(result.RawValue << integerPart);
if (neg)
{
result = FP.One / result;
}
return(result);
}
/// <summary>
/// Returns the base-2 logarithm of a specified number.
/// Provides at least 9 decimals of accuracy.
/// </summary>
/// <exception cref="ArgumentOutOfRangeException">
/// The argument was non-positive
/// </exception>
internal static FP Log2(FP x)
{
if (x.RawValue <= 0)
{
throw new ArgumentOutOfRangeException("Non-positive value passed to Ln", "x");
}
// This implementation is based on Clay. S. Turner's fast binary logarithm
// algorithm (C. S. Turner, "A Fast Binary Logarithm Algorithm", IEEE Signal
// Processing Mag., pp. 124,140, Sep. 2010.)
long b = 1U << (FP.FRACTIONAL_PLACES - 1);
long y = 0;
long rawX = x.RawValue;
while (rawX < FP.ONE)
{
rawX <<= 1;
y -= FP.ONE;
}
while (rawX >= (FP.ONE << 1))
{
rawX >>= 1;
y += FP.ONE;
}
var z = FP.FromRaw(rawX);
for (int i = 0; i < FP.FRACTIONAL_PLACES; i++)
{
z = FP.FastMul(z, z);
if (z.RawValue >= (FP.ONE << 1))
{
z = FP.FromRaw(z.RawValue >> 1);
y += b;
}
b >>= 1;
}
return(FP.FromRaw(y));
}
/// <summary>
/// Returns the natural logarithm of a specified number.
/// Provides at least 7 decimals of accuracy.
/// </summary>
/// <exception cref="ArgumentOutOfRangeException">
/// The argument was non-positive
/// </exception>
public static FP Ln(FP x)
{
return(FP.FastMul(Log2(x), FP.Ln2));
}